The utilization of non-periodic functions for telecommunication signaling may utilize waveforms derived from functions in either of the two equivalent forms:
                                                        f              g                        ⁡                          (              t              )                                ⁢                      ⅇ                          t              ·                              cos                ⁡                                  (                                                            2                                              1                        -                        g                                                              ⁢                    π                                    )                                                              ⁢                      ⅇ                          ⅈ              ·              t              ·                              sin                ⁡                                  (                                                            2                                              1                        -                        g                                                              ⁢                    π                                    )                                                                    ⁢                                  ⁢        or                            (        1        )                                                      f            g                    ⁡                      (            t            )                          =                  ⅇ                      t            ⁢                                                  ⁢                          ⅈ              ⁡                              (                                  2                                      2                    -                    g                                                  )                                                                        (        2        )            
In these equations, i is the imaginary constant equal to √{square root over (−1)}, t is the time parameter, and g has the effect of varying the geometry of the waveform where g=2 corresponds to a complex circle, as the above reduce to the Euler term eti. Known techniques such as Quadrature Amplitude Modulation or QAM are based on complex circles. Values of g>2 correspond to complex spirals of increasingly rapid growth and increasingly lower frequency.
There are other variations of these waveforms based on altering parameters including but not limited to amplitude, frequency, phase, signal duration and direction of rotation. There is also a unified way of specifying many possible variations of waveforms via the generalized spiral formula:
                                          f            g                    ⁡                      (            t            )                          =                              [                                          κ                0                            ⁢                              ⅇ                                  ⅈ                                      ω                    0                                                                        ]                    ⁢                      ⅇ                                          [                                                      κ                    1                                    ⁢                                      ⅇ                                          ⅈω                      1                                                                      ]                            ⁢                              (                                  t                  +                                      t                    0                                                  )                            ⁢                              ⅈ                                                      [                                                                  κ                        2                                            ⁢                                              ⅇ                                                  ⅈ                          ⁢                                                                                                          ⁢                                                      ω                            2                                                                                                                ]                                    ⁢                                      (                                          2                                              2                        -                        g                                                              )                                                                                                          (        3        )            
Dropping the square brackets for conciseness, the general spiral formula (Equation 3) may be written as:
                                          f            g                    ⁡                      (            t            )                          =                              κ            0                    ⁢                      ⅇ                          ⅈ                              ω                0                                              ⁢                      ⅇ                                          κ                1                            ⁢                                                ⅇ                                      ⅈω                    1                                                  ⁡                                  (                                      t                    +                                          t                      0                                                        )                                            ⁢                              ⅈ                                                      κ                    2                                    ⁢                                                            ⅇ                                              ⅈ                        ⁢                                                                                                  ⁢                                                  ω                          2                                                                                      ⁡                                          (                                              2                                                  2                          -                          g                                                                    )                                                                                                                              (        4        )            
The scope of this formula may be characterized by the effect of varying its parameters on the set of symbol waveforms that may be generated. The effects of varying the parameters include amplitude modulation that utilizes multiple values of κ0, phase modulation that utilize multiple values of ω0, time reversal that utilize κ1=±1, or equivalently ω1=0 and ω1=a, frequency modulation that has a scale κ1, time shifts that utilize multiple values of t0, rotational reversal that utilize κ2=±1, or equivalently ω2=0 and ω2=π, waveform shape modulation that vary g with higher values of g that correspond to more rapid growth and lower frequency and other variations with general values of κ2, e1, and ω2.
For the case of g=2, corresponding to the standard telecommunications conditions of no amplitude growth, the only parameters that are usually modified are amplitude, frequency and phase. By contrast, for signals where g>2, the generalized spiral formula shows that there are additional parameters, or degrees of freedom, available for encoding information in the signal. Of particular practical significance are time reversal, rotational reversal and waveform shape modulation.
The effect of these three additional degrees of freedom is to dramatically increase the number of possible modulation sets for a particular alphabet size. By way of example, consider a communications channel with an alphabet size of M=8, such that three bits are encoded per symbol. For non-spiral based communications, utilizing the standard technique of superposition of the I & Q waveforms, the possible modulation sets are shown in Table 1.
TABLE 1Non-spiral modulation sets for an alphabet of 8PhasesAmplitudesFrequenciesSymbols8118421841282228241821481818142812481188
Thus one could encode the alphabet utilizing eight phases, one amplitude and one frequency (e.g. what is generally known as 8-PSK), or one could utilize four phases, two amplitudes and one frequency which if it was utilized practically would be referred to as 8-QAM and so on. As shown in Table 1 there are just ten possible modulation sets for non-spiral modulation for M=8.
For values of g>2, one may utilize time reversal to double the number of symbols, and then also utilize rotation reversal to double the number of symbols again. Thus for a particular value of growth g, the possible modulation sets are shown in Table 2.
TABLE 2Spiral modulation sets for an alphabet of 8 and just one growth value.PhasesAmplitudesFrequenciesRotationTimeSymbols811008421008412008222008241008214008181008142008124008118008811008421008412008411108411018222008241008214008221108221018212108212018211118181008142008124008118008141108141018121118114108114018112118
As shown in Table 2, there are approximately 32 different modulation sets for spiral-based communications with an 8-symbol alphabet. However, this represents approximately 32 possible combinations for just a single value of growth “g”. Practical values of “g” vary from approximately 2.1 to 3.0, with noticeable differences being observable at steps of approximately 0.05. I.e., a symbol set with a g value of approximately 2.60 will exhibit noticeably different properties when compared to a symbol set with a g value of approximately 2.65. As such there are approximately [2.1 3.0 0.05]=18 useful values of g, giving an overall number of modulation combinations for an 8-symbol alphabet of approximately 32*18=576. This is 576/10=58 times more combinations when compared to non-spiral communications given an 8-symbol alphabet.
For larger alphabet sizes such as approximately 64, the number of possible modulation sets becomes enormous for spiral based modulation. While the mathematics behind spiral based communications is complex, the implementation need not be. For the transmitter, the result of the mathematics may be a simple lookup table which is indexed by symbol number and time. These values are sent to a Digital-Analog Converter or DAC whose output feeds into a radio frequency or RF stage which is typically either a mixer or directly into a power amplifier. The symbols are transmitted across the communications channel, and a version of the symbols generally corrupted by channel conditions is digitized at the receiver. The digitized symbols may then be fed through a series of matched filters in order to determine the received symbol.
It may be desirable for a communication system to use different modulation sets at different times, either due to changing channel conditions or for enhanced security. From an implementation perspective, a spiral based transceiver may evaluate at run time the complex equations necessary for synthesizing the transmitter look-up table/receiver matched filter coefficients and thus generate the necessary data on the fly, or have the data for the various modulation sets computed off line and the results simply stored in the memory system.
For a typical application with an alphabet size of approximately 64, and approximately 32 samples per symbol, the transmitter look-up table includes just 64*32=2048 values, with each value being typically represented by approximately 16 bits, giving a table size of approximately 4096 bytes per modulation set. Thus in typical memory system sizes available today, it is possible to store thousands of possible modulation sets and thus the latter approach of pre-computing modulation set data makes more sense.
A further benefit of storing thousands of modulation sets in memory is that it permits modulation sets to be instantaneously switched. This results in significant benefits as will be described. To comprehend the value of switching modulation, it is helpful to know some empirical results of investigations into spiral based modulation. Different spiral modulation combinations have markedly different characteristics in the presence of various channel impairments. For example, some modulation combinations work better with Additive White Gaussian Noise or AWGN, whereas others perform better in the presence of coherent noise. For a given channel impairment (e.g. coherent noise), there is a subset of the possible modulation combinations that perform optimally, and an even larger subset of the possible modulation combinations that perform acceptably. Some modulation combinations impose higher demands upon the underlying hardware than other combinations. For example, modulation combinations with higher g values typically require power amplifiers that may slew more rapidly and concomitantly analog to digital converters or ADC that may sample the signal more rapidly. Since spiral based signals are not periodic, the Nyquist sampling theorem does not apply and thus the band-limited received signal must be measured at multiple points in order to determine its shape. In general, the performance of spiral based communications may be arbitrarily improved by increasing the number of samples taken at the receiver. However, as the sample rate increases, so too do the demands on the ADC and subsequent digital filters. Thus it is possible to trade off communications performance with hardware cost and the amount of power required to run the ADC and subsequent digital filters. This has enormous implications for certain types of communications applications, as will be described.
It should be apparent that spiral based communications offers tremendous flexibility when compared to traditional forms of communications. While this flexibility may be exploited in the design of any given communications channel, it is particularly beneficial when implemented in a software defined radio or SDR. In a SDR, some or all of the features that traditionally were implemented in hardware are instead implemented in software such as non-transitory storage media. Typically this includes functions such as mixing, demodulation, modulation, filtering, phase locking and so on. Indeed, a state of the art SDR includes a powerful microprocessor, high speed ADC, high speed DAC, a power amplifier and an antenna. All of the radio functions are implemented in the microprocessor's non-transitory storage media. As such, an SDR is the perfect vehicle for exploiting the flexibility of spiral based communications.